The minimization problem with double well profile (REF ) as well as the associated Euler-Lagrange equation (REF ) are classical examples exhibiting topological soliton or kink solutions, with typical examples being the Allen-Cahn (or quartic) or sine-Gordon equations for which an analytic formula for the solution is available. Under mild regularity conditions on the GSFE functional \(\Phi \) , we have the following classical result (see e.g. [1]}, Prop. 2.1):
